The issue of optimal and market allocation of economic activity across space has always been a central issue in economic theory from the seminal works of Hotelling (1929) and Salop (1979). Many authors have already investigated in depth the existence (and sometimes the non-existence) of optimal and/or market allocation in static models (among them, Starrett, 1974 and 1978). Recently, some authors have tried to tackle the issue of the spatial allocation of economic activity in dynamic frameworks, namely within economic growth frameworks (thus, with capital accumulation).
This research line, initiated by Brito (2004), is nicely surveyed by Desmet and Rossi-Hansberg (2010). Two aspects turn out to be crucial: factor mobility and space modelling. On the first aspect, Brito and Boucekkine, Camacho and Zou (2009) consider frictionless capital mobility while Brock and Xepapadeas (2008) use the trick of a spatial externality to model the spatial component (without capital mobility). In the former papers, the production function at any place is neoclassical (decreasing returns in capital): capital flow from regions with low marginal productivity of capital to regions with high marginal productivity.
Concerning spatial modelling, both Brito and Boucekkine et al. consider an infinite spatial line, symmetrical to the infinite time horizon adopted in standard growth theory. But while Boucekkine et al. introduce spatial discounting (again mimicking time discounting) to ensure the convergence of the resulting utilitarian objective function, Brito postulates a non-utilitarian objective function and gets rid of spatial discounting. Brock and Xepapadeas (2008) take a compact interval to model space, therefore avoiding this technical problem.
A striking finding of Boucekkine et al. is that the model with infinite time and space support, and utilitarian objective function with strictly concave preferences, which is admittedly the most natural candidate for spatial Ramsey model, suffers from a structural ill-posedness problem: in contrast to the non-spatial Ramsey model, the initial value of the co-state variable is no longer sufficient to determine the optimal paths. Rather than considering a non-standard objective function as in Brito (2004), we follow here the example of Brock and Xepapadeas (2008) and consider a more realistic compact modelling of space. However, in sharp contrast to these authors, we do not consider a compact interval on the real line but a circle, which is another traditional modelling of space in economics (Salop, 1979). A very important advantage of our modelling is that it does not require the specification of any space boundary conditions, to which the solution paths are arguably extremely sensitive. We also keep capital mobility in line with Boucekkine et al. (2009).
In order to gather at least some partial clean analytical results, we will consider an AK technology. Using recent developments in dynamic programming in infinitely-dimensioned problems (Bensoussan et al., 2007), previously implemented in the literature of vintage capital theory (Fabbri and Gozzi, 2008), it is possible to identify an explicit value function under quite reasonable parametric conditions. As in the standard AK model, a benevolent social planner will choose a constant consumption level (that is detrended consumption): in our spatial set-up, this turns out to be true not only over time but also over space assuming homogeneous preferences and technology over space.
Also, we will show that the aggregate capital stock, that’s the aggregation of capital stocks across space, lies on a balanced growth path from t = 0, featuring the absence of transitional dynamics for the aggregate variables as in the standard AK theory. A major and striking departure from the standard theory will nonetheless emerge: while the aggregate capital stock shows no dynamics, an intense reallocation of capital will take place across space giving rise to transition dynamics at any point in space. Even more strikingly, these spatio-temporal dynamics will lead to the convergence of (time detrended) capital stocks across space to the same common value whatever the initial spatial distribution of capital under certain realistic assumptions.
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